HEOR Core Metrics: OR vs HR Intuition Guide

Establishing the underlying logic from Probability to Odds to Hazard Velocity in Health Technology Assessment

Author

Xiaoge Zhang, PhD

Published

June 1, 2026

1 Introduction

In health technology assessment (HTA) submissions, clinical evidence is summarized using various relative treatment effects. The choice of these metrics is not arbitrary; it dictates how clinical efficacy is translated into survival curves, state occupancy traces, and ultimately cost-effectiveness results (QALYs and ICERs).

Two of the most frequently used—and often misunderstood—relative metrics are:

Odds Ratio (OR): A static, cross-sectional measure comparing the odds of an event.
Hazard Ratio (HR): A dynamic, time-to-event measure comparing the instantaneous rate of event occurrence.

This guide provides the mathematical derivations, statistical properties, R implementations, and strategic HTA implications of these two core metrics.

2 Survival Functions Derivation: From Hazard to Survival

To understand the Hazard Ratio, we must first understand the relationship between the instantaneous Hazard Rate \(h(t)\), Cumulative Hazard \(H(t)\), and the Survival Function \(S(t)\).

2.1 Mathematical Definitions

Let \(T \ge 0\) represent the random variable for time-to-event.¹

¹ Denotes survival time, measuring the duration from trial baseline to the event of interest. In parametric survival modeling, this random variable \(T\) is assumed to follow distributions like the Weibull, Log-Normal, or Gompertz.

Survival Function \(S(t)\): The probability of surviving (i.e., not experiencing the event) beyond time \(t\): \[S(t) = P(T > t)\]
Cumulative Distribution Function \(F(t)\): The probability of experiencing the event by time \(t\): \[F(t) = P(T \le t) = 1 - S(t)\]
Probability Density Function \(f(t)\): The rate of event occurrence per unit of time: \[f(t) = \frac{d}{dt}F(t) = \frac{d}{dt}(1 - S(t)) = -S'(t)\]
Hazard Rate \(h(t)\): The instantaneous failure rate at time \(t\), conditional on survival up to \(t\): \[h(t) = \lim_{\Delta t \to 0} \frac{P(t \le T < t + \Delta t \mid T \ge t)}{\Delta t} = \frac{f(t)}{S(t)}\]

2.2 Step-by-Step Derivation of \(S(t)\)

Step 1: Express \(h(t)\) in terms of \(S(t)\) and its derivative

Substitute \(f(t) = -S'(t)\) into the definition of \(h(t)\): \[h(t) = \frac{-S'(t)}{S(t)}\]

Step 2: Recognize the derivative of the natural logarithm

By the chain rule: \[\frac{d}{dt}\ln(S(t)) = \frac{S'(t)}{S(t)}\] Substituting this back into Step 1 yields: \[h(t) = -\frac{d}{dt}\ln(S(t))\]

Step 3: Integrate both sides

Integrate both sides from \(0\) to \(t\): \[\int_0^t h(u) du = \int_0^t -\frac{d}{du}\ln(S(u)) du\] The left side is defined as the Cumulative Hazard Function \(H(t)\): \[H(t) = \int_0^t h(u) du\] The right side can be evaluated using the Fundamental Theorem of Calculus: \[\int_0^t -\frac{d}{du}\ln(S(u)) du = [-\ln(S(u))]_0^t = -\ln(S(t)) - (-\ln(S(0)))\]

Step 4: Apply boundary conditions

At time \(t=0\), the probability of survival is \(1\), so \(S(0) = 1\). Since \(\ln(1) = 0\): \[H(t) = -\ln(S(t))\]

Step 5: Solve for \(S(t)\)

Multiply by \(-1\) and take the exponential of both sides: \[S(t) = e^{-H(t)} = \exp\left(-\int_0^t h(u) du\right)\]

3 Odds Ratio (OR): The Static Endpoint Snapshot

The Odds Ratio evaluates the relative likelihood of a binary outcome. It is widely used in HTA for endpoints like Complete Response (CR) rates, Adverse Event (AE) rates, or any cross-sectional indicator.

3.1 Mathematical Formulation

Probability (\(p\)): The proportion of times an event occurs (\(p \in [0, 1]\)). In a time-to-event context, if we define the binary event as experiencing the outcome (e.g., progression or death) by a specific time \(t\), this probability \(p\) is equivalent to the Cumulative Distribution Function \(F(t) = 1 - S(t)\) defined in the previous section.
Odds: The ratio of the probability of the event occurring to the probability of it not occurring: \[\text{Odds} = \frac{p}{1 - p}\]
Odds Ratio (OR): The ratio of the odds of the event in the treatment group to the odds of the event in the control group: \[\text{OR} = \frac{\text{Odds}_{\text{T}}}{\text{Odds}_{\text{C}}} = \frac{p_{\text{T}} / (1 - p_{\text{T}})}{p_{\text{C}} / (1 - p_{\text{C}})}\]

3.2 Logit Link and Logistic Regression

In evidence synthesis (such as Bayesian Meta-Analysis), relative treatment effects are modeled on the log-odds scale using the logit link function: \[\text{logit}(p) = \ln\left(\frac{p}{1 - p}\right) = \beta_0 + \beta_1 X\] Under this model: \[\ln(\text{OR}) = \text{logit}(p_{\text{T}}) - \text{logit}(p_{\text{C}}) = \beta_1\] \[\text{OR} = e^{\beta_1}\]

Strategic Rationale in HTA

Symmetry: The log-odds scale maps probabilities from \([0, 1]\) to \((-\infty, +\infty)\), preventing out-of-bounds predictions.
Appraisal Defence: Under a logistic framework, the OR remains mathematically stable across different study baseline risk levels, unlike the Relative Risk (RR), which is structurally constrained by the baseline rate (e.g., if control \(p_{\text{C}} = 0.6\), RR cannot exceed \(1.67\), but OR has no such restriction).

3.3 R Implementation: Calculating OR via Logistic Regression

To ground the mathematical definition, we simulate a Phase III randomized controlled trial evaluating a binary landmark endpoint, such as the Complete Response (CR) rate at 24 weeks. The trial consists of N = 500 patients randomized 1:1, with an assumed true response probability of 40% in the Control arm (p_C = 0.40) and 55% in the active Treatment arm (p_T = 0.55). Below is the R implementation to simulate this cohort, fit the logistic regression model, and calculate the Odds Ratio.

Code

library(dplyr)
library(knitr)

# Generate synthetic binary response data (e.g., Complete Response)
set.seed(2)
n_patients <- 500
data_binary <- data.frame(
  Treatment = c(rep("Control", n_patients/2), rep("Drug A", n_patients/2))
) %>%
  mutate(
    # Control has ~40% CR rate; Drug A has ~55% CR rate
    CR_prob = ifelse(Treatment == "Control", 0.40, 0.55),
    CR_achieved = rbinom(n(), 1, CR_prob)
  )

# 1. Direct cross-tabulation
table_cr <- table(data_binary$Treatment, data_binary$CR_achieved)
colnames(table_cr) <- c("No CR", "CR")

# 2. Fit Logistic Regression
fit_logistic <- glm(CR_achieved ~ Treatment, data = data_binary, family = binomial)

# 3. Extract log-odds and convert to Odds Ratio
summary_coef <- summary(fit_logistic)$coefficients
log_or <- summary_coef["TreatmentDrug A", "Estimate"]
se_log_or <- summary_coef["TreatmentDrug A", "Std. Error"]

or_est <- exp(log_or)
or_ci_lower <- exp(log_or - 1.96 * se_log_or)
or_ci_upper <- exp(log_or + 1.96 * se_log_or)

true_or <- (0.55 / 0.45) / (0.40 / 0.60)
true_log_or <- log(true_or)

results_df <- data.frame(
  Parameter = c("Log-Odds (Beta)", "Odds Ratio (OR)"),
  `True Parameter` = c(round(true_log_or, 4), round(true_or, 3)),
  Estimate = c(round(log_or, 4), round(or_est, 3)),
  `Std. Error` = c(round(se_log_or, 4), "-"),
  `95% CI` = c(
    sprintf("%.4f to %.4f", log_or - 1.96 * se_log_or, log_or + 1.96 * se_log_or),
    sprintf("%.3f to %.3f", or_ci_lower, or_ci_upper)
  ),
  check.names = FALSE
)

# 4. Display results as styled tables
kable(table_cr, format = "html", table.attr = "class='table table-striped table-hover' style='width: 50%; margin-bottom: 20px;'", caption = "Complete Response (CR) Contingency Table")

Complete Response (CR) Contingency Table
	No CR	CR
Control	153	97
Drug A	117	133

Code

kable(results_df, format = "html", table.attr = "class='table table-striped table-hover' style='width: 70%;'", caption = "Logistic Regression Model Results")

Logistic Regression Model Results
Parameter	True Parameter	Estimate	Std. Error	95% CI
Log-Odds (Beta)	0.6061	0.5839	0.1814	0.2283 to 0.9395
Odds Ratio (OR)	1.8330	1.7930	-	1.257 to 2.559

4 Hazard Ratio (HR): The Dynamic Instantaneous Velocity

The Hazard Ratio is the standard relative metric for time-to-event outcomes, such as Overall Survival (OS) or Progression-Free Survival (PFS).

The Cox Proportional Hazards (PH) framework is a semi-parametric regression model widely used in survival analysis to relate the survival time of patients to one or more predictor variables. By modeling the hazard rate as a function of covariates without specifying the functional form of the baseline hazard, it offers a flexible and robust way to estimate relative treatment effects.

4.1 The Proportional Hazards (PH) Assumption

Under the Cox Proportional Hazards framework, the hazard for an individual with covariate vector \(X\) is: \[h(t \mid X) = h_0(t) e^{\beta X}\] Where \(h_0(t)\) is the baseline hazard function. The Hazard Ratio comparing treatment (\(X=1\)) to control (\(X=0\)) is: \[\text{HR}(t) = \frac{h(t \mid X=1)}{h(t \mid X=0)} = \frac{h_0(t) e^{\beta}}{h_0(t)} = e^{\beta}\] Since the baseline hazard \(h_0(t)\) cancels out, the HR is assumed to be constant over time.

4.2 Compatibility of Parametric Distributions with PH

In parametric survival analysis, only specific distributions are mathematically compatible with a constant Hazard Ratio:

Strictly PH-Compatible Distributions:
- Exponential Distribution: Hazard rate \(h(t) = \lambda\) is constant. The ratio is \(\lambda_{\text{T}} / \lambda_{\text{C}}\), which is naturally constant.
- Weibull Distribution: Hazard rate \(h(t) = \lambda \alpha t^{\alpha - 1}\). If we assume a shared shape parameter \(\alpha\) between arms, the hazard ratio is \(\lambda_{\text{T}} / \lambda_{\text{C}}\), which is constant.
- Gompertz Distribution: Hazard rate \(h(t) = a e^{b t}\). If different arms share the growth rate \(b\), the HR is \(a_{\text{T}} / a_{\text{C}}\), which is constant.
- Restricted Cubic Splines (on Log-Cumulative Hazard Scale / Royston-Parmar Model): These flexible parametric models use cubic splines (smooth polynomials joined at user-specified “knots”) to model the log-cumulative hazard function. If all spline parameters (representing the shape of the baseline log-cumulative hazard) are shared across treatment arms, and the treatment effect is modeled as a simple additive constant on the log-cumulative hazard scale (shifting only the intercept), the model enforces a constant Hazard Ratio. This allows for highly flexible baseline hazard shapes (e.g., bathtub or peaked shapes) while strictly maintaining the Proportional Hazards assumption.
Non-PH Distributions (Accelerated Failure Time / AFT):
- Log-Normal, Log-Logistic, and Generalized Gamma: These assume that the treatment multiplies survival time (time acceleration/deceleration) rather than scaling the hazard. Under these distributions, the implied HR changes continuously over time, violating the Cox PH assumption.

4.3 R Implementation: Calculating HR via Cox Regression

Here is how to calculate and interpret the Hazard Ratio in R using synthetic survival data.

Code

library(survival)
library(knitr)

# Generate synthetic survival data
set.seed(18)
n_patients <- 200
data_survival <- data.frame(
  Treatment = c(rep(0, n_patients/2), rep(1, n_patients/2)) # 0: Control, 1: Drug A
) %>%
  mutate(
    # Generate Weibull-distributed survival times
    # Control: shape = 1.2, scale = 0.04 (median OS ~ 24 months)
    # Drug A: shape = 1.2, scale = 0.02 (median OS ~ 43 months; HR = 0.5)
    true_scale = ifelse(Treatment == 0, 0.04, 0.02),
    u = runif(n()),
    time = (-log(u) / true_scale)^(1/1.2),
    # Add random censoring before 60 months
    censor_time = runif(n(), 10, 80),
    observed_time = pmin(time, censor_time, 60),
    event = ifelse(time <= observed_time, 1, 0)
  )

# Fit Cox Proportional Hazards Model
fit_cox <- coxph(Surv(observed_time, event) ~ Treatment, data = data_survival)

# Extract and convert coefficients
summary_cox <- summary(fit_cox)
log_hr <- summary_cox$coefficients["Treatment", "coef"]
se_log_hr <- summary_cox$coefficients["Treatment", "se(coef)"]

hr_est <- exp(log_hr)
hr_ci_lower <- summary_cox$conf.int["Treatment", "lower .95"]
hr_ci_upper <- summary_cox$conf.int["Treatment", "upper .95"]

# True parameters: HR = 0.5, log(HR) = log(0.5) = -0.6931
true_hr <- 0.5
true_log_hr <- log(true_hr)

# Display results as a styled table
results_cox_df <- data.frame(
  Parameter = c("Log-Hazard (Beta)", "Hazard Ratio (HR)"),
  `True Parameter` = c(round(true_log_hr, 4), round(true_hr, 3)),
  Estimate = c(round(log_hr, 4), round(hr_est, 3)),
  `Std. Error` = c(round(se_log_hr, 4), "-"),
  `95% CI` = c(
    sprintf("%.4f to %.4f", log_hr - 1.96 * se_log_hr, log_hr + 1.96 * se_log_hr),
    sprintf("%.3f to %.3f", hr_ci_lower, hr_ci_upper)
  ),
  check.names = FALSE
)
kable(results_cox_df, format = "html", table.attr = "class='table table-striped table-hover' style='width: 70%;'", caption = "Cox Proportional Hazards Model Results")

Cox Proportional Hazards Model Results
Parameter	True Parameter	Estimate	Std. Error	95% CI
Log-Hazard (Beta)	-0.6931	-0.6937	0.1628	-1.0127 to -0.3747
Hazard Ratio (HR)	0.5000	0.5000	-	0.363 to 0.687

5 Methodological Note: Key Differences & HTA Misconceptions

While both Odds Ratios (OR) and Hazard Ratios (HR) express relative treatment effects, they operate on different scales, represent different data structures, and are mathematically distinct.

5.1 Key Conceptual Differences

Static vs. Dynamic:
- Odds Ratio (OR) is a static, cross-sectional measure at a specific landmark time point (e.g., Complete Response rate at 24 weeks). It ignores when events occurred or the velocity of their occurrence during the study.
- Hazard Ratio (HR) is a dynamic, time-to-event measure that compares the instantaneous rate of event occurrence across the entire follow-up period.
Mathematical Definition:
- OR represents the ratio of the odds of an event occurring (i.e., \(p / (1 - p)\)).
- HR represents the ratio of the instantaneous hazard rates (i.e., \(h(t \mid X=1) / h(t \mid X=0)\)).

5.2 Common HTA Modeling Pitfalls

Avoid Direct Equivalency: A common error in indirect treatment comparisons (ITCs) is treating an OR for a binary landmark survival endpoint (e.g., proportion of patients alive at 1 year) as a constant Hazard Ratio. Because the cumulative event rate changes over time, the relationship between the cumulative event OR and the underlying HR is non-linear and dynamic.
Survivor Bias: Applying an event-based OR (such as response or relapse rates) directly as a hazard multiplier in partitioned survival models or state transition models without accounting for baseline hazards introduces selection bias.
Effect Magnification: For common events (e.g., response rates around 50%), the OR is numerically more extreme (further from 1.0) than the Relative Risk (RR) or Hazard Ratio (HR), which can lead to overestimating treatment efficacy if misinterpreted.

6 Conclusion & Strategic Considerations

When drafting or auditing HTA submissions, ensure statistical coherence by following these guidelines:

Check scale consistency: Ensure all Weibull or spline parameters derived from trial analysis match the model cycle length (e.g., converting annual scale parameters to monthly).
Verify AFT distributions: If using Log-Normal or Log-Logistic curves, do not report a single, constant HR in the submission text, as it varies by design.
Avoid OR-to-HR conversions: Never use a constant conversion multiplier to convert binary ORs into time-to-event HRs.
Expose the ROI: Clearly demonstrate to the appraisal committee whether the drug’s value is driven by “entry” (Odds Ratio of response) or “durability” (Hazard Ratio of progression/relapse).

Technical Library Node: 04_NMA/05_NMA_Technical_Prep/11_HR_OR_Intuition.qmd
Methodological Alignment: NICE Decision Support Unit (DSU) TSD 19 & 21.